Math Doubts

Power Rule of Exponents

Formula

${(b^{\displaystyle \, m})}^{\displaystyle n} \,=\, b^{\displaystyle \, mn}$

The power of an exponent with a base is equal to the product of the exponents with same base, is called the power rule of exponents.

Introduction

$b$, $m$ and $n$ are constants. The literals $b$ and $m$ formed an exponential term $b^{\displaystyle \,m}$. The literal $n$ is a power for exponential term $b^{\displaystyle \,m}$ and formed a special term ${(b^{\displaystyle \, m})}^{\displaystyle n}$ in exponential form.

The power $n$ of an exponent $m$ with base $b$ is equal to the product of the exponents $m$ and $n$ with base $b$.

${(b^{\displaystyle \, m})}^{\displaystyle n} \,=\, b^{\displaystyle \, mn}$

It is called as the power rule for exponents and used as a formula when a quantity is power of an exponential term.

Proof

Learn how to derive the power rule of exponents in algebraic form with understandable process.

Verification

${(2^3)}^4$ is a term in which an exponential term $2^3$ has $4$ as its power..

$\implies$ ${(2^3)}^4 \,=\, {(2 \times 2 \times 2)}^4$

$\implies$ ${(2^3)}^4$ $\,=\,$ $(2 \times 2 \times 2)$ $\times$ $(2 \times 2 \times 2)$ $\times$ $(2 \times 2 \times 2)$ $\times$ $(2 \times 2 \times 2)$

$\implies$ ${(2^3)}^4$ $\,=\,$ $2 \times 2 \times 2$ $\times$ $2 \times 2 \times 2$ $\times$ $2 \times 2 \times 2$ $\times$ $2 \times 2 \times 2$

$\implies$ ${(2^3)}^4$ $\,=\,$ $2^{12}$

In this example, $3$ and $4$ are exponents in the left-hand side of the equation but $12$ is power in the right-hand side of the equation. The number $12$ can be factored as the product of the numbers $3$ and $4$.

$\,\,\, \therefore \,\,\,\,\,\,$ ${(2^3)}^4$ $\,=\,$ $2^{\, 3 \times 4}$

Therefore, it is verified that the power of an exponential term is equal to the product of the exponents with same base.

Math Questions

The math problems with solutions to learn how to solve a problem.

Learn solutions

Math Worksheets

The math worksheets with answers for your practice with examples.

Practice now

Math Videos

The math videos tutorials with visual graphics to learn every concept.

Watch now

Subscribe us

Get the latest math updates from the Math Doubts by subscribing us.

Learn more

Math Doubts

A free math education service for students to learn every math concept easily, for teachers to teach mathematics understandably and for mathematicians to share their maths researching projects.

Copyright © 2012 - 2023 Math Doubts, All Rights Reserved